The fine structure of the singular set of area-minimizing integral currents II: Rectifiability of flat singular points with singularity degree larger than $1$

MR image reconstruction from partial k-space using singularity function representation

Let f be a complex valued function from the open upper halfplane E of the complex plane. We study the set of all z∈∂E such that there exist two Stoltz angles V 1 , V 2 in E with vertices in z ( i.e. , V i is a closed angle with vertex at z and V i \{ z } ⊂ E , i = 1, 2) such that the function f has different cluster sets with respect to these angles at z . E. P. Dolzhenko showed that this set of singular points is G ∂σ and σ-porous for every f . He posed the question of whether each G ∂σ σ-porous set is a set of such singular points for some f . We answer this question negatively. Namely, we construct a G ∂ porous set, which is a set of such singular points for no function f .

We study the dynamics of the action of an automaton group on the set of infinite words, and more precisely the discontinuous points of the map which associates to a point its set of stabilizers — the singular points. We show that, for any Mealy automaton, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group. We prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. We also provide some conditions for such examples to exist. Finally, we study some dynamical properties of the Schreier graphs in the boundary.

Medical image denoising using one-dimensional singularity function model

Developing A.D. Aleksandrov’s ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a C-submanifold in a smooth Riemannian manifold. Let Y1 be a two-dimensional compact connected C-submanifold with non-empty boundary in some smooth twodimensional Riemannian manifold (X, g) without boundary. Let us consider the intrinsic metric (the infimum of the lengths of paths, connecting a pair of points.) of the interior IntY1 of Y1, and extend it by continuity (operation lim ) to the boundary points of ∂Y1. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines ∂Y1 up to isometry of ambient space (X, g). We also consider the case dimYj = dimX = n, n > 2.

We review recent probabilistic results on covariant Schrodinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian manifolds. 1 A review of covariant Schrodinger operators on smooth Riemannian manifolds Let us start by taking a look at covariant Schrodinger-type operators on Riemannian manifolds: Assume that E →M is a smooth finite dimensional Hermitian vector bundle over a possibly noncompact smooth Riemannian manifold M without boundary, with a Hermitian covariant derivative ∇ on E, which means that ∇ is a complex linear map ∇ : ΓC∞(M,E) −→ ΩC∞(M,E), such that for all Ψ1,Ψ2 ∈ ΓC∞(M,E) one has: d(Ψ1,Ψ2) = (∇Ψ1,Ψ2) + (Ψ1,∇Ψ2). (1) Then the symmetric nonnegative sesquilinear form

The problem of defining the singular points of quasi-linear differential-algebraic systems

In the process of mechanism design and trajectory planning of in-parallel actuated manipulators, it is necessary to grasp the locations of all of singular points in the working space. This paper proposes a new method for numerically searching for the singular points of in-parallel actuated manipulators based on the pressure angle. A singular-point curve is defined as a set of singular points. Since the working space is divided into multiple regions by the singular-point curves, the end-effector cannot move freely from one region to any other regions. Kinematic and static characteristics at singular points are analyzed by pressure angles and instantaneous screw axes. Singular points of the Stewart-Platform are investigated based on the proposed method. Characteristics in the neighborhoods of singular points are analyzed from the points of view of output error, isotropy of output force and driving force, and the practical working space is defined.

Geodesic is a kind of shortest path among all curves in a metric space, which originally appeared in the Gaussian period. Since then, it was extensively applied in various branches of mathematics and physics, such as Riemannian geometry, digital geometry, Einstein’s relativity, etc. This thesis mainly discusses geodesic definitions in Euclidean space and those in smooth manifolds after introducing the basic theory of smooth manifolds. At first, the thesis applies three ways to define geodesics on a surface, including the geodesic curvature method, the shortest distance method, and the relation of the principal normal of a curve and the normal vector of a surface. Then, the paper introduces some basic concepts in Riemannian manifolds. Finally, it gives a general definition of geodesics in a smooth Riemannian manifold.

We study the parabolic obstacle problem $$\begin{aligned} \Delta u-u_t=f\chi _{\{u>0\}}, \quad u\ge 0,\quad f\in L^p \quad \text{ with }\quad f(0)=1 \end{aligned}$$ and obtain two monotonicity formulae, one that applies for general free boundary points and one for singular free boundary points. These are used to prove a second order Taylor expansion at singular points (under a pointwise Dini condition), with an estimate of the error (under a pointwise double Dini condition). Moreover, under the assumption that $$f$$ is Dini continuous, we prove that the set of regular points is locally a (parabolic) $$C^1$$ -surface and that the set of singular points is locally contained in a union of (parabolic) $$C^1$$ manifolds.

In the process of mechanism design and trajectory planning of in-parallel actuated manipulators, it is necessary to know the locations of all singular points in the working space. We propose a new method for numerically locating the singular points of in-parallel actuated manipulators based on the pressure angle. A singular-point curve is defined as a set of singular points. Kinematic and static characteristics at singular points are analyzed by pressure angles and instantaneous screw axes. Singular points of the Stewart Platform are investigated using the proposed method. The neighborhood of singular points is determined from output error, isotropy of output force and driving force, and thus the practical working space is defined.

The boundary behavior of finitely bi-Lipschitz mappings on smooth Riemannian manifolds is studied.

Differential-algebraic power system models exhibit singularity induced (SI) bifurcations as the parameters vary. The SI bifurcation point, which occurs when the system equilibria meet the singularity of the algebraic part of the model, belongs to a large set of other singular points called a singular set. At any singular point, the DAE model breaks down; meaning a vector field on a set satisfying algebraic part of the DAE model is not defined. Thus, DAE's cannot be reduced to a set of ordinary differential equations (ODEs) at the singular points. In terms of power system dynamics, at the singular points, relationship between generator angles and load bus variables breaks down and the DAE model cannot predict system behavior. Thus, singular points impose a dynamic constraint to the system model and computational tools are essential to locate them. We propose a novel method to compute singular points of the DAE model for the multi-machine power systems. The identification of the singular points is formulated as a bifurcation problem of a set of algebraic equations whose parameters are the generator angles. The method implements Newton-Raphson (NR)/Newton-Raphson-Seydel (NRS) methods to compute singular points. The proposed method is implemented into Voltage Stability Toolbox (VST) and simulations results are presented for several power system examples. Singular points are depicted together with the system equilibria to visualize static and dynamic limits.

This paper investigates qualitative features of singular points of differential-algebraic power system model. Large-scale dynamical systems such as power systems are generally modeled as a set of parameter dependent differential-algebraic equations (DAEs). As the parameter changes, the DAE model of a power system may exhibit a singularity-induced (SI) bifurcation as well as other frequently encountered bifurcations such as saddle node and Hopf types. The SI bifurcation point, which occurs when system equilibria meet the singularity of the algebraic part of the model, belongs to a large set of other singular points called a singular set. At any singular point in this set, the DAE model breaks down, meaning a vector field on a set satisfying algebraic part of the DAE model (known as constraint manifold) is not well defined. Thus, DAE's cannot be reduced to a set of ordinary differential equations (ODE's) at the singular points and ODE solvers fail to converge in a neighborhood of a singular point. The singular points split into two complementary classes: Those from which exactly two distinct trajectories emanate (repelling) and those at which exactly two distinct solutions terminate (attracting). In order to predict system trajectories originating from a point close to a singular point, the qualitative properties of singular points at any given parameter value are identified and simulation results are presented for a 3 bus test system.

In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input-multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class.